A LevelFurther MathsEdexcelFurther Pure 1Revision NotesFurther VectorsFurther VectorsShortest Distances using Vector Rules

Shortest Distances using Vector Rules (Edexcel A Level Further Maths: Further Pure 1): Revision Note

Exam code: 9FM0

Written by: Mark Curtis

Updated on 20 January 2026

Shortest distances using vector rules

How do I find the shortest distance between a point and a plane?

The shortest (perpendicular) distance of the point P with coordinates $(α, β, γ)$ from the plane with equation $n_{1} x + n_{2} y + n_{3} z + d = 0$ is given by the formula
- $\frac{|n_{1} α + n_{2} β + n_{3} γ + d|}{\sqrt{{n_{1}}^{2} + {n_{2}}^{2} + {n_{3}}^{2}}}$

A plane with equation n_1 x+n_2 y+n_3 z+d=0 and the point P with position vector p from an origin O with coordinates (alpha, beta, gamma). P is above the plane and the shortest distance between P and the plane is shown as a dotted arrow perpendicular to the plane.

Examiner Tips and Tricks

This formula is given in the formula booklet.

How do I find the shortest distance between a point and a line?

The shortest (perpendicular) distance of the point P (with position vector $p$ ) from the line with equation $r = a + λ b$ through A (with position vector $a$ ) is given by the formula
- $\frac{|(p - a) \times b|}{|b|}$

A line with equation r=a+lambda*b where A is a point on the line with position vector 'a' shown from an origin O and direction vector b shown on the line. The point P is above the line with position vector p shown from O. The shortest distance is shown by a dotted arrow which is perpendicular to the direction of the line,

Examiner Tips and Tricks

You must learn this formula as it is not given in the formula booklet.

How do I find the shortest distance between two skew lines?

The shortest (perpendicular) distance between two skew lines, $r = a_{1} + λ b_{1}$ and $r = a_{2} + μ b_{2}$ , is given by the formula
- $\frac{|(a_{1} - a_{2}) \cdot (b_{1} \times b_{2})|}{|b_{1} \times b_{2}|}$

Two skew lines shown. The lower has equation r=a_1 + lambda*b_1 which passes through the point A_1 (with position vector a_1 shown from an origin) and goes in direction b_1. The upper has equation r=a_2 + mu*b_2 which passes through the point A_2 (with position vector a_2 shown from an origin) and goes in direction b_2, The shortest distance between the skew lines is shown as a dotted arrow which is perpendicular to the directions of both lines.

Examiner Tips and Tricks

You must learn this formula as it is not given in the formula booklet.

How do I find the shortest distance between two parallel planes?

The shortest (perpendicular) distance between two parallel planes is the same as finding the shortest distance from a point to a plane, as follows:
- Take any point P on one of the planes
- then use the formula given above for the shortest distance of the point P with coordinates $(α, β, γ)$ to the other plane $n_{1} x + n_{2} y + n_{3} z + d = 0$
  - $\frac{|n_{1} α + n_{2} β + n_{3} γ + d|}{\sqrt{{n_{1}}^{2} + {n_{2}}^{2} + {n_{3}}^{2}}}$

How do I find the shortest distance between a plane and a line parallel to the plane?

The shortest (perpendicular) distance between a plane and a line that is parallel to the plane is the same as finding the shortest distance from a point to a plane, as follows:
- Take any point P on the line
- then use the formula given above for the shortest distance of the point P with coordinates $(α, β, γ)$ to the plane $n_{1} x + n_{2} y + n_{3} z + d = 0$
  - $\frac{|n_{1} α + n_{2} β + n_{3} γ + d|}{\sqrt{{n_{1}}^{2} + {n_{2}}^{2} + {n_{3}}^{2}}}$

How do I find the shortest distance between two parallel lines?

The shortest (perpendicular) distance between two parallel lines is the same as finding the shortest distance from a point to a line, as follows:
- Take any point P on one of the lines
- then use the formula given above for the shortest distance of P (with direction vector $p$ ) to the other line (with equation $r = a + λ b$ )
  - $\frac{|(p - a) \times b|}{|b|}$

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Shortest Distances using Vector Rules (Edexcel A Level Further Maths: Further Pure 1): Revision Note

Shortest distances using vector rules

How do I find the shortest distance between a point and a plane?

How do I find the shortest distance between a point and a line?

How do I find the shortest distance between two skew lines?

How do I find the shortest distance between two parallel planes?

How do I find the shortest distance between a plane and a line parallel to the plane?

How do I find the shortest distance between two parallel lines?

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Mark Curtis